Polynomial invariants of classical subgroups of $\operatorname{GL}_{2}$: Conjugation over finite fields

TL;DR

This paper investigates the structure of polynomial invariants under conjugation actions of classical subgroups of GL(2) over finite fields, revealing that these invariant rings are either polynomial rings or hypersurfaces.

Contribution

It extends the understanding of invariant rings from infinite fields to finite fields for classical subgroups of GL(2), identifying their algebraic structures.

Findings

Invariant rings over finite fields are either polynomial rings or hypersurfaces.

Explicit descriptions of invariants for subgroups like traceless and symmetric matrices.

Comparison with known results over infinite fields.

Abstract

Consider the conjugation action of the general linear group $\operatorname{GL}_{2}(K)$ on the polynomial ring $K[X_{2 \times 2}]$. When $K$ is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the determinant. We describe the ring of invariants when $K$ is a finite field, and show that it is a hypersurface. We also consider the other classical subgroups, and the polynomial rings corresponding to other subspaces of matrices such as the traceless and symmetric matrices. In each case, we show that the invariant ring is either a polynomial ring or a hypersurface.